Scale-invariant quantum anomalous Hall effect in magnetic topological insulators beyond the two-dimensional limit

Antiferromagnetic quantum anomalous Hall effect under spin flips and flops

The interplay between nontrivial band topology and layered antiferromagnetism in MnBi2Te4 has opened a new avenue for exploring topological phases of matter1-4. The quantum anomalous Hall effect5 and axion insulator state6 have been observed in odd and even number layers of MnBi2Te4, and the quantum metric nonlinear Hall effect7,8 has been shown to exist in this topological antiferromagnet. The rich and complex antiferromagnetic spin dynamics in MnBi2Te4 is expected to generate new quantum anomalous Hall phenomena that are absent in conventional ferromagnetic topological insulators, but experimental observations are still unknown. Here we fabricate a device of 7-septuple-layer MnBi2Te4 covered with an AlOx capping layer, which enables the investigation of antiferromagnetic quantum anomalous Hall effect over wide parameter spaces. By tuning the gate voltage and perpendicular magnetic field, we uncover a cascade of quantum phase transitions that can be attributed to the influence of complex spin configurations on edge state transport. Furthermore, we find that an in-plane magnetic field enhances both the coercive field and the exchange gap of the surface state, in contrast to that in the ferromagnetic quantum anomalous Hall state. Combined with numerical simulations, we propose that these peculiar features arise from the spin flip and flop transitions that are inherent to a van der Waals antiferromagnet. The versatile tunability of the quantum anomalous Hall effect in MnBi2Te4 paves the way for potential applications in topological antiferromagnetic spintronics9,10.

Cryogenic in-memory computing using magnetic topological insulators

Machine learning algorithms have proven to be effective for essential quantum computation tasks such as quantum error correction and quantum control. Efficient hardware implementation of these algorithms at cryogenic temperatures is essential. Here we utilize magnetic topological insulators as memristors (termed magnetic topological memristors) and introduce a cryogenic in-memory computing scheme based on the coexistence of a chiral edge state and a topological surface state. The memristive switching and reading of the giant anomalous Hall effect exhibit high energy efficiency, high stability and low stochasticity. We achieve high accuracy in a proof-of-concept classification task using four magnetic topological memristors. Furthermore, our algorithm-level and circuit-level simulations of large-scale neural networks demonstrate software-level accuracy and lower energy consumption for image recognition and quantum state preparation compared with existing magnetic memristor and complementary metal-oxide-semiconductor technologies. Our results not only showcase a new application of chiral edge states but also may inspire further topological quantum-physics-based novel computing schemes.

Controllable magnetism and an anomalous Hall effect in (Bi1-xSbx)2Te3-intercalated MnBi2Te4 multilayers

MnBi2Te4-based superlattices not only enrich the materials family of magnetic topological insulators, but also offer a platform for tailoring magnetic properties and interlayer magnetic coupling through the strategic insertion layer design. Here, we present the electrical and magnetic characterization of (Bi1-xSbx)2Te3-intercalated MnBi2Te4 multilayers grown by molecular beam epitaxy. By precisely adjusting the Sb-to-Bi ratio in the spacer layer, the magneto-transport response is modulated, unveiling the critical role of Fermi level tuning in optimizing the anomalous Hall signal and reconfiguring the magnetic ground state. Moreover, by varying the interlayer thickness, tunable magnetic coupling is achieved, enabling precise control over ferromagnetic and antiferromagnetic components. These findings pave the way for the exploration of versatile magnetic topological phases in quantum materials systems.

Robust Topological Interface States in a Lateral Magnetic-Topological Heterostructure

Introducing uniform magnetic order in two-dimensional (2D) topological insulators by constructing heterostructures of TI and magnet is a promising way to realize the high-temperature Quantum Anomalous Hall effect. However, the topological properties of 2D materials are susceptible to several factors that make them difficult to maintain, and whether topological interface states (TISs) can exist at magnetic-topological heterostructure interfaces is largely unknown. Here, it is experimentally shown that TISs in a lateral heterostructure of CrTe2/Bi(110) are robust against disorder, defects, high magnetic fields (time-reversal symmetry-breaking perturbations), and elevated temperature (77 K). The lateral heterostructure is realized by lateral epitaxial growth of bilayer (BL) Bi to monolayer CrTe2 grown on graphite. Scanning Tunneling Microscopy and non-contact Atomic Force Microscopy demonstrate a black phosphorus-like structure with low atomic buckling (less than 0.1 Å) of the BL Bi(110), indicating the presence of its topological properties. Scanning tunneling spectroscopy and energy-dependent dI/dV mapping further confirm the existence of topologically induced one-dimensional in-gap states localized at the interface. These results demonstrate the robustness of TISs in lateral magnetic-topological heterostructures, which is competitive with those in vertically stacked magnetic-topological heterostructures and provides a promising route for constructing planar high-density non-dissipative devices using TISs.

Direct observation of chiral edge current at zero magnetic field in a magnetic topological insulator

The chiral edge current is the boundary manifestation of the Chern number of a quantum anomalous Hall (QAH) insulator. The van der Waals antiferromagnet MnBi2Te4 is theorized to be a QAH in odd-layers but has shown Hall resistivity below the quantization value at zero magnetic field. Here, we perform scanning superconducting quantum interference device (sSQUID) microscopy on these seemingly failed QAH insulators to image their current distribution. When gated to the charge neutral point, our device exhibits edge current, which flows unidirectionally on the odd-layer boundary both with vacuum and with the even-layers. The edge current chirality reverses with the magnetization of the bulk. Surprisingly, we find the edge channels coexist with finite bulk conduction even though the bulk chemical potential is in the band gap, suggesting their robustness under significant edge-bulk scattering. Our result establishes the existence of chiral edge currents in a topological antiferromagnet and offers an alternative for identifying QAH states.

Multi-level chiral edge states in Janus M2XS2Se2 (M = V, Ti; X = W, Mo) monolayers with high Curie temperature and sizable nontrivial topological gaps

Quantum anomalous Hall (QAH) insulators with dissipation-less chiral edge channels provide ideal platforms for the exploration of topological materials and low-power spintronic devices. However, the ultralow operation temperature and small nontrivial gaps are the bottlenecks for QAH insulators towards future applications. Here, a new family of QAH insulators, that is, Janus M2XS2Se2 (M = V, Ti; X = W, Mo) monolayers, are proposed to be ferromagnets with large perpendicular magnetic anisotropy (PMA) and high Curie temperature above room temperature. Moreover, the present M2XS2Se2 monolayers hold sizable nontrivial topological gaps, resulting in the 1st chiral edge state with Chern number C = -1. Unexpectedly, there also exists an occupied 2nd chiral edge state below the Fermi level. Although all M2XS2Se2 monolayers retain their PMA characteristics on application of biaxial strain, various topological phase transitions are present. The V2WS2Se2 monolayer preserves the QAH state regardless of strain, while the V2MoS2Se2 and Ti2WS2Se2 monolayers transform from QAH states to metallic states under tensile strains. The present M2XS2Se2 monolayers show competitive advantages among the reported materials for the development of topological electronic devices.

Spin-Hall conductivity and optical characteristics of noncentrosymmetric quantum spin Hall insulators: the case of PbBiI

Quantum spin Hall insulators have attracted significant attention in recent years. Understanding the optical properties and spin Hall effect in these materials is crucial for technological advancements. In this study, we present theoretical analyses to explore the optical properties, Berry curvature and spin Hall conductivity of pristine and perturbed PbBiI using the linear combination of atomic orbitals and the Kubo formula. The system is not centrosymmetric and it is hosting at the same time Rashba spin-splitting and quantized spin Hall conductivity. Our calculations reveal that the electronic structure can be modified using staggered exchange fields and electric fields, leading to changes in the optical properties. Additionally, the spin Berry curvature and spin Hall conductivity are investigated as a function of the energy and temperature. The results indicate that due to the small dynamical spin Hall conductivity, generating an ac spin current in the PbBiI requires the use of external magnetic fields or magnetic materials.

Atomic-Scale Characterization of Dilute Dopants in Topological Insulators via STEM-EDS Using Registration and Cell Averaging Techniques

Magnetic dopants in three-dimensional topological insulators (TIs) offer a promising avenue for realizing the quantum anomalous Hall effect (QAHE) without the necessity for an external magnetic field. Understanding the relationship between site occupancy of magnetic dopant elements and their effect on macroscopic property is crucial for controlling the QAHE. By combining atomic-scale energy-dispersive X-ray spectroscopy (EDS) maps obtained by aberration-corrected scanning transmission electron microscopy (AC-STEM) and novel data processing methodologies, including semi-automatic lattice averaging and frame registration, we have determined the substitutional sites of Mn atoms within the 1.2% Mn-doped Sb2Te3 crystal. More importantly, the methodology developed in this study extends beyond Mn-doped Sb2Te3 to other quantum materials, traditional semiconductors, and even electron irradiation sensitive materials.

Giant Hall Switching by Surface-State-Mediated Spin-Orbit Torque in a Hard Ferromagnetic Topological Insulator

Topological insulators (TI) and magnetic topological insulators (MTI) can apply highly efficient spin-orbit torque (SOT) and manipulate the magnetization with their unique topological surface states (TSS) with ultrahigh efficiency. Here, efficient SOT switching of a hard MTI, V-doped (Bi,Sb)2Te3 (VBST), with a large coercive field that can prevent the influence of an external magnetic field, is demonstrated. A giant switched anomalous Hall resistance of 9.2 kΩ is realized, among the largest of all SOT systems, which makes the Hall channel a good readout and eliminates the need to fabricate complicated magnetic tunnel junction (MTJ) structures. The SOT switching current density can be reduced to 2.8 × 105 A cm-2, indicating its high efficiency. Moreover, as the Fermi level is moved away from the Dirac point by both gate and composition tuning, VBST exhibits a transition from edge-state-mediated to surface-state-mediated transport, thus enhancing the SOT effective field to (1.56 ± 0.12) × 10-6 T A-1 cm2 and the interfacial charge-to-spin conversion efficiency to 3.9 ± 0.3 nm-1. The findings establish VBST as an extraordinary candidate for energy-efficient magnetic memory devices.

Atomic-Layer-Controlled Magnetic Orders in MnBi2Te4-Bi2Te3 Topological Heterostructures

The natural van der Waals superlattice MnBi2Te4-(Bi2Te3)m provides an optimal platform to combine topology and magnetism in one system with minimal structural disorder. Here, we show that this system can harbor both ferromagnetic (FM) and antiferromagnetic (AFM) orders and that these magnetic orders can be controlled in two different ways by either varying the Mn-Mn distance while keeping the Bi2Te3/MnBi2Te4 ratio constant or vice versa. We achieve this by creating atomically engineered sandwich structures composed of Bi2Te3 and MnBi2Te4 layers. We show that the AFM order is exclusively determined by the Mn-Mn distance, whereas the FM order depends only on the overall Bi2Te3/MnBi2Te4 ratio regardless of the distance between the MnBi2Te4 layers. Our results shed light on the origins of the AFM and FM orders and provide insights into how to manipulate magnetic orders not only for the MnBi2Te4-Bi2Te3 system but also for other magneto-topological materials.

Half-quantum mirror Hall effect

We predict a half-quantized mirror Hall effect induced by mirror symmetry in strong topological insulator films. These films are known to host a pair of gapless Dirac cones in the first Brillouin zone associated with surface electrons. Our findings reveal that mirror symmetry assigns a unique mirror parity to each Dirac cone, resulting in a half-quantized Hall conductance of ± e 2 2 h for each cone. Despite the total electrical Hall conductance being null due to time-reversal invariance, the difference in the Hall conductance between the two cones yields a quantized Hall conductance ofe 2 h for the difference in mirror currents. The effect of helical edge mirror current - a crucial feature of this quantum effect - may, in principle, be determined by means of electrical measurements. The half-quantum mirror Hall effect reveals a type of mirror-symmetry induced quantum anomaly in a time-reversal invariant lattice system, giving rise to a topological metallic state of matter with time-reversal invariance.

Spin Gapless Quantum Materials and Devices

Quantum materials, with nontrivial quantum phenomena and mechanisms, promise efficient quantum technologies with enhanced functionalities. Quantum technology is held back because a gap between fundamental science and its implementation is not fully understood yet. In order to capitalize the quantum advantage, a new perspective is required to figure out and close this gap. In this review, spin gapless quantum materials, featured by fully spin-polarized bands and the electron/hole transport, are discussed from the perspective of fundamental understanding and device applications. Spin gapless quantum materials can be simulated by minimal two-band models and could help to understand band structure engineering in various topological quantum materials discovered so far. It is explicitly highlighted that various types of spin gapless band dispersion are fundamental ingredients to understand quantum anomalous Hall effect. Based on conventional transport in the bulk and topological transport on the boundaries, various spintronic device aspects of spin gapless quantum materials as well as their advantages in different models for topological field effect transistors are reviewed.

Even-Odd Layer-Dependent Exchange Bias Effect in MnBi2Te4 Chern Insulator Devices

Magnetic topological materials with coexisting magnetism and nontrivial band structures exhibit many novel quantum phenomena, including the quantum anomalous Hall effect, the axion insulator state, and the Weyl semimetal phase. As a stoichiometric layered antiferromagnetic topological insulator, thin films of MnBi2Te4 show fascinating even-odd layer-dependent physics. In this work, we fabricate a series of thin-flake MnBi2Te4 devices using stencil masks and observe the Chern insulator state at high magnetic fields. Upon magnetic field training, a large exchange bias effect is observed in odd but not in even septuple layer (SL) devices. Through theoretical calculations, we attribute the even-odd layer-dependent exchange bias effect to the contrasting surface and bulk magnetic properties of MnBi2Te4 devices. Our findings reveal the microscopic magnetic configuration of MnBi2Te4 thin flakes and highlight the challenges in replicating the zero magnetic field quantum anomalous Hall effect in odd SL MnBi2Te4 devices.

Inducing superconductivity in quantum anomalous Hall regime

Interfacing the quantum anomalous Hall insulator with a conventional superconductor is known to be a promising manner for realizing a topological superconductor, which has been continuously pursued for years. Such a proximity route depends to a great extent on the control of the delicate interfacial coupling of the two constituents. However, a recent experiment reported the failure to reproduce such a topological superconductor, which is ascribed to the negligence of the electrical short by the superconductor in the theoretical proposal. Here, we reproduce this topological superconductor with attention to the interface control. The resulted conductance matrix under a wide magnetic field range agrees with the fingerprint of this topological superconductor. This allows us to develop a phase diagram that unveils three regions parameterized by various coupling limits, which not only supports the feasibility to fabricate the topological superconductor by proximity but also fully explains the origin of the previous debate. The present work provides a comprehensible guide on fabricating the topological superconductor.

Engineering Plateau Phase Transition in Quantum Anomalous Hall Multilayers

The plateau phase transition in quantum anomalous Hall (QAH) insulators corresponds to a quantum state wherein a single magnetic domain gives way to multiple domains and then reconverges back to a single magnetic domain. The layer structure of the sample provides an external knob for adjusting the Chern number C of the QAH insulators. Here, we employ molecular beam epitaxy to grow magnetic topological insulator multilayers and realize the magnetic field-driven plateau phase transition between two QAH states with odd Chern number change ΔC. We find that critical exponents extracted for the plateau phase transitions with ΔC = 1 and ΔC = 3 in QAH insulators are nearly identical. We construct a four-layer Chalker-Coddington network model to understand the consistent critical exponents for the plateau phase transitions with ΔC = 1 and ΔC = 3. This work will motivate further investigations into the critical behaviors of plateau phase transitions with different ΔC in QAH insulators.

On the topological surface states of the intrinsic magnetic topological insulator Mn-Bi-Te family

We review recent progress in the electronic structure study of intrinsic magnetic topological insulators (MnBi2Te4) · (Bi2Te3)n ([Formula: see text]) family. Specifically, we focus on the ubiquitously (nearly) gapless behavior of the topological Dirac surface state observed by photoemission spectroscopy, even though a large Dirac gap is expected because of surface ferromagnetic order. The dichotomy between experiment and theory concerning this gap behavior is perhaps the most critical and puzzling question in this frontier. We discuss various proposals accounting for the lack of magnetic effect on the topological Dirac surface state, which are mainly categorized into two pictures, magnetic reconfiguration and topological surface state redistribution. Band engineering towards opening a magnetic gap of topological surface states provides great opportunities to realize quantized topological transport and axion electrodynamics at higher temperatures.

Epitaxial Growth and Characterization of Nanoscale Magnetic Topological Insulators: Cr-Doped (Bi0.4Sb0.6)2Te3

The exploration initiated by the discovery of the topological insulator (BixSb1-x)2Te3 has extended to unlock the potential of quantum anomalous Hall effects (QAHEs), marking a revolutionary era for topological quantum devices, low-power electronics, and spintronic applications. In this study, we present the epitaxial growth of Cr-doped (Bi0.4Sb0.6)2Te3 (Cr:BST) thin films via molecular beam epitaxy, incorporating various Cr doping concentrations with varying Cr/Sb ratios (0.025, 0.05, 0.075, and 0.1). High-quality crystalline of the Cr:BST thin films deposited on a c-plane sapphire substrate has been rigorously confirmed through reflection high-energy electron diffraction (RHEED), X-ray diffraction (XRD), and high-resolution transmission electron microscopy (HRTEM) analyses. The existence of a Cr dopant has been identified with a reduction in the lattice parameter of BST from 30.53 ± 0.05 to 30.06 ± 0.04 Å confirmed by X-ray diffraction, and the valence state of Cr verified by X-ray photoemission (XPS) at binding energies of ~573.1 and ~583.5 eV. Additionally, the influence of Cr doping on lattice vibration was qualitatively examined by Raman spectroscopy, revealing a blue shift in peaks with increased Cr concentration. Surface characteristics, crucial for the functionality of topological insulators, were explored via Atomic Force Microscopy (AFM), illustrating a sevenfold reduction in surface roughness as the Cr concentration increased from 0 to 0.1. The ferromagnetic properties of Cr:BST were examined by a superconducting quantum interference device (SQUID) with a magnetic field applied in out-of-plane and in-plane directions. The Cr:BST samples exhibited a Curie temperature (Tc) above 50 K, accompanied by increased magnetization and coercivity with increasing Cr doping levels. The introduction of the Cr dopant induces a transition from n-type ((Bi0.4Sb0.6)2Te3) to p-type (Cr:(Bi0.4Sb0.6)2Te3) carriers, demonstrating a remarkable suppression of carrier density up to one order of magnitude, concurrently enhancing carrier mobility up to a factor of 5. This pivotal outcome is poised to significantly influence the development of QAHE studies and spintronic applications.

Electrical switching of the edge current chirality in quantum anomalous Hall insulators

A quantum anomalous Hall (QAH) insulator is a topological phase in which the interior is insulating but electrical current flows along the edges of the sample in either a clockwise or counterclockwise direction, as dictated by the spontaneous magnetization orientation. Such a chiral edge current eliminates any backscattering, giving rise to quantized Hall resistance and zero longitudinal resistance. Here we fabricate mesoscopic QAH sandwich Hall bar devices and succeed in switching the edge current chirality through thermally assisted spin-orbit torque (SOT). The well-quantized QAH states before and after SOT switching with opposite edge current chiralities are demonstrated through four- and three-terminal measurements. We show that the SOT responsible for magnetization switching can be generated by both surface and bulk carriers. Our results further our understanding of the interplay between magnetism and topological states and usher in an easy and instantaneous method to manipulate the QAH state.

3D Quantum Anomalous Hall Effect in Magnetic Topological Insulator Trilayers of Hundred-Nanometer Thickness

Magnetic topological states refer to a class of exotic phases in magnetic materials with the non-trivial topological property determined by magnetic spin configurations. An example of such states is the quantum anomalous Hall (QAH) state, which is a zero magnetic field manifestation of the quantum Hall effect. Current research in this direction focuses on QAH insulators with a thickness of less than 10 nm. Here, molecular beam epitaxy (MBE) is employed to synthesize magnetic TI trilayers with a thickness of up to ≈106 nm. It is found that these samples exhibit well-quantized Hall resistance and vanishing longitudinal resistance at zero magnetic field. By varying the magnetic dopants, gate voltages, temperature, and external magnetic fields, the properties of these thick QAH insulators are examined and the robustness of the 3D QAH effect is demonstrated. The realization of the well-quantized 3D QAH effect indicates that the nonchiral side surface states of the thick magnetic TI trilayers are gapped and thus do not affect the QAH quantization. The 3D QAH insulators of hundred-nanometer thickness provide a promising platform for the exploration of fundamental physics, including axion physics and image magnetic monopole, and the advancement of electronic and spintronic devices to circumvent Moore's law.

Insight into the quantum anomalous Hall states in two-dimensional kagome Cr3Se4 and Fe3S4 monolayers

To realize the quantum anomalous Hall (QAH) effect in two-dimensional (2D) intrinsic magnetic materials, which combines insulating bulk states and metallic edge channel states, is still challenging in experiment. Here, based on first-principles calculations, we predicted two stable kagome-latticed QAH insulators: Cr3Se4 and Fe3S4 monolayers, with the Chern number C = 1. It is found that both structures exhibit a large magnetic anisotropy energy and sizable band gaps, and a topological phase transition from C = -1 to C = 1 occurs when the magnetization orientation changes from the z-axis to the -z-axis. Remarkably, the non-trivial topological properties are robust against biaxial strains of up to ±6%. Furthermore, a variable high Chern number of C = 2 or C = 3 can be observed by stacking two or three layers of the QAH monolayer with an MoS2 insulator. Our results signify that such layered kagome materials can be promising platforms for exploring novel QAH physics.

Axion insulator state in hundred-nanometer-thick magnetic topological insulator sandwich heterostructures

An axion insulator is a three-dimensional (3D) topological insulator (TI), in which the bulk maintains the time-reversal symmetry or inversion symmetry but the surface states are gapped by surface magnetization. The axion insulator state has been observed in molecular beam epitaxy (MBE)-grown magnetically doped TI sandwiches and exfoliated intrinsic magnetic TI MnBi2Te4 flakes with an even number layer. All these samples have a thickness of ~ 10 nm, near the 2D-to-3D boundary. The coupling between the top and bottom surface states in thin samples may hinder the observation of quantized topological magnetoelectric response. Here, we employ MBE to synthesize magnetic TI sandwich heterostructures and find that the axion insulator state persists in a 3D sample with a thickness of ~ 106 nm. Our transport results show that the axion insulator state starts to emerge when the thickness of the middle undoped TI layer is greater than ~ 3 nm. The 3D hundred-nanometer-thick axion insulator provides a promising platform for the exploration of the topological magnetoelectric effect and other emergent magnetic topological states, such as the high-order TI phase.

Antiferromagnetic topological insulator with selectively gapped Dirac cones

Antiferromagnetic (AF) topological materials offer a fertile ground to explore a variety of quantum phenomena such as axion magnetoelectric dynamics and chiral Majorana fermions. To realize such intriguing states, it is essential to establish a direct link between electronic states and topology in the AF phase, whereas this has been challenging because of the lack of a suitable materials platform. Here we report the experimental realization of the AF topological-insulator phase in NdBi. By using micro-focused angle-resolved photoemission spectroscopy, we discovered contrasting surface electronic states for two types of AF domains; the surface having the out-of-plane component in the AF-ordering vector displays Dirac-cone states with a gigantic energy gap, whereas the surface parallel to the AF-ordering vector hosts gapless Dirac states despite the time-reversal-symmetry breaking. The present results establish an essential role of combined symmetry to protect massless Dirac fermions under the presence of AF order and widen opportunities to realize exotic phenomena utilizing AF topological materials.

Strain-tunable magnetism and topological states in layered VBi2Te4

Similar to the magnetic topological insulator of MnBi2Te4, recent studies have demonstrated that VBi2Te4 is also an ideal candidate to explore many intriguing quantum states. Different from the strong interlayer antiferromagnetic (AFM) coupling in layered MnBi2Te4, based on first-principles calculations, we find that the energy difference between AFM and ferromagnetic (FM) orders in layered VBi2Te4 is much smaller than that of MnBi2Te4. Specifically, it is found that the interlayer FM coupling can be readily achieved by applying strain. Further electronic band structures reveal that the VBi2Te4 bilayer is a time-reversal symmetry broken quantum spin Hall insulator with a spin Chern number of CS = 1, which is essentially different from the QAH state with a Chern number of C = 1 in the MnBi2Te4 bilayer. Most strikingly, the topological states of the magnetic VBi2Te4 bilayer can be well tuned by strain, whose topological phase diagram is mapped out as a function of strain by employing continuum model analyses. All of these results indicate that the layered VBi2Te4 not only enriches the family of magnetic topological materials, but also provides a promising platform to explore more exotic quantum phenomena.

Manipulating Topological Phases in Magnetic Topological Insulators

Magnetic topological insulators (MTIs) are a group of materials that feature topological band structures with concurrent magnetism, which can offer new opportunities for technological advancements in various applications, such as spintronics and quantum computing. The combination of topology and magnetism introduces a rich spectrum of topological phases in MTIs, which can be controllably manipulated by tuning material parameters such as doping profiles, interfacial proximity effect, or external conditions such as pressure and electric field. In this paper, we first review the mainstream MTI material platforms where the quantum anomalous Hall effect can be achieved, along with other exotic topological phases in MTIs. We then focus on highlighting recent developments in modulating topological properties in MTI with finite-size limit, pressure, electric field, and magnetic proximity effect. The manipulation of topological phases in MTIs provides an exciting avenue for advancing both fundamental research and practical applications. As this field continues to develop, further investigations into the interplay between topology and magnetism in MTIs will undoubtedly pave the way for innovative breakthroughs in the fundamental understanding of topological physics as well as practical applications.

Quantized resistance revealed at the criticality of the quantum anomalous Hall phase transitions

In multilayered magnetic topological insulator structures, magnetization reversal processes can drive topological phase transitions between quantum anomalous Hall, axion insulator, and normal insulator states. Here we report an examination of the critical behavior of two such transitions: the quantum anomalous Hall to normal insulator (QAH-NI), and quantum anomalous Hall to axion insulator (QAH-AXI) transitions. By introducing a new analysis protocol wherein temperature dependent variations in the magnetic coercivity are accounted for, the critical behavior of the QAH-NI and QAH-AXI transitions are evaluated over a wide range of temperature and magnetic field. Despite the uniqueness of these different transitions, quantized longitudinal resistance and Hall conductance are observed at criticality in both cases. Furthermore, critical exponents were extracted for QAH-AXI transitions occurring at magnetization reversals of two different magnetic layers. The observation of consistent critical exponents and resistances in each case, independent of the magnetic layer details, demonstrates critical behaviors in quantum anomalous Hall transitions to be of electronic rather than magnetic origin. Our finding offers a new avenue for studies of phase transition and criticality in QAH insulators.

Controlling the Zero Hall Plateau in a Quantum Anomalous Hall Insulator by In-Plane Magnetic Field

We investigate the quantum anomalous Hall plateau transition in the presence of independent out-of-plane and in-plane magnetic fields. The perpendicular coercive field, zero Hall plateau width, and peak resistance value can all be systematically controlled by the in-plane magnetic field. The traces taken at various fields almost collapse into a single curve when the field vector is renormalized to an angle as a geometric parameter. These results can be explained consistently by the competition between magnetic anisotropy and in-plane Zeeman field, and the close relationship between quantum transport and magnetic domain structure. The accurate control of zero Hall plateau facilitates the search for chiral Majorana modes based on the quantum anomalous Hall system in proximity to a superconductor.

Intrinsic and extrinsic dopings in epitaxial films MnBi2Te4

The intrinsic antiferromagnetic topological insulator MnBi₂Te₄and members of its family have been the subject of theoretical and experimental research, which has revealed the presence of a variety of defects and disorders that are crucial in determining the topological and magnetic properties. This also brings about challenges in realizing the quantum states like the quantum anomalous Hall and the axion insulator states. Here, utilizing cryogenic magnetoelectric transport and magnetic measurements, we systematically investigate the effects arising from intrinsic doping by antisite defects and extrinsic doping by Sb in MnBi₂Te₄epitaxial films grown by molecular beam epitaxy. We demonstrate that the nonequilibrium condition in epitaxy allows a wide growth window for optimizing the crystalline quality and defect engineering. While the intrinsic antisite defects caused by the intermixing between Bi and Mn can be utilized to tune the Fermi level position as evidenced by a p-to-n conductivity transition, the extrinsic Sb-doping not only compensates for this doping effect but also modifies the magnetism and topology of the film, during which a topological phase transition is developed. Conflicting reports from the theoretical calculations and experimental measurements in bulk crystals versus epitaxial films are addressed, which highlights the intimate correlation between the magnetism and topology as well as the balance between the Fermi-level positioning and defect control. The present study provides an experimental support for the epitaxial growth of the intrinsic topological insulator and underlines that the topology, magnetism, and defect engineering should be revisited for enabling a steady and reliable film production.

Evolution of Dopant-Concentration-Induced Magnetic Exchange Interaction in Topological Insulator Thin Films

To date, the quantum anomalous Hall effect has been realized in chromium (Cr)- and/or vanadium(V)-doped topological insulator (Bi,Sb)₂Te₃ thin films. In this work, we use molecular beam epitaxy to synthesize both V- and Cr-doped Bi₂Te₃ thin films with controlled dopant concentration. By performing magneto-transport measurements, we find that both systems show an unusual yet similar ferromagnetic response with respect to magnetic dopant concentration; specifically the Curie temperature does not increase monotonically but shows a local maximum at a critical dopant concentration. We attribute this unusual ferromagnetic response observed in Cr/V-doped Bi₂Te₃ thin films to the dopant-concentration-induced magnetic exchange interaction, which displays evolution from van Vleck-type ferromagnetism in a nontrivial magnetic topological insulator to Ruderman-Kittel-Kasuya-Yosida (RKKY)-type ferromagnetism in a trivial diluted magnetic semiconductor. Our work provides insights into the ferromagnetic properties of magnetically doped topological insulator thin films and facilitates the pursuit of high-temperature quantum anomalous Hall effect.

Floquet Engineering of Nonequilibrium Valley-Polarized Quantum Anomalous Hall Effect with Tunable Chern Number

Here, we propose that Floquet engineering offers a strategy to realize the nonequilibrium quantum anomalous Hall effect (QAHE) with tunable Chern number. Using first-principles calculations and Floquet theorem, we unveil that QAHE related to valley polarization (VP-QAHE) is formed from the hybridization of Floquet sidebands in the two-dimensional family MSi₂Z₄ (M = Mo, W, V; Z = N, P, As) by irradiating circularly polarized light (CPL). Through the tuning of frequency, intensity, and handedness of CPL, the Chern number of VP-QAHE is highly tunable and up to C = ±4, which attributes to light-induced trigonal warping and multiple-band inversion at different valleys. The chiral edge states and quantized plateau of Hall conductance are visible inside the global band gap, thereby facilitating the experimental measurement. Our work not only establishes Floquet engineering of nonequilibrium VP-QAHE with tunable Chern number in realistic materials but also provides an avenue to explore emergent topological phases under light irradiation.

Electrically Tunable Topological Phase Transition in van der Waals Heterostructures

The realization and control of the quantum anomalous Hall (QAH) effect are highly desirable for the development of spintronic and quantum devices. In this work, we propose a van der Waals (vdW) heterostructure of ultrathin MnBi₂Se₄ and Bi₂Se₃ layers and demonstrate that it is an excellent tunable QAH platform by using model Hamiltonian and density functional theory simulations. Its band gap closes and reopens as external electric field increases, manifesting a novel topological phase transition with an electric field of ∼0.06 V/Å. This heterostructure has other major advantageous, such as large topological band gap, perpendicular magnetization, and strong ferromagnetic ordering. Our work provides clear physical insights and suggests a new strategy for experimental realization and control of the QAH effect in real materials.

Electrical Manipulation of Topological Phases in a Quantum Anomalous Hall Insulator

Quantum anomalous Hall phases arising from the inverted band topology in magnetically doped topological insulators have emerged as an important subject of research for quantization at zero magnetic fields. Though necessary for practical implementation, sophisticated electrical control of molecular beam epitaxy (MBE)-grown quantum anomalous Hall matter have been stymied by growth and fabrication challenges. Here, a novel procedure is demonstrated, employing a combination of thin-film deposition and 2D material stacking techniques, to create dual-gated devices of the MBE-grown quantum anomalous Hall insulator, Cr-doped (Bi,Sb)₂ Te₃ . In these devices, orthogonal control over the field-induced charge density and the electric displacement field is demonstrated. A thorough examination of material responses to tuning along each control axis is presented, realizing magnetic property control along the former and a novel capability to manipulate the surface exchange gap along the latter. Through electrically addressing the exchange gap, the capabilities to either strengthen the quantum anomalous Hall state or suppress it entirely and drive a topological phase transition to a trivial state are demonstrated. The experimental result is explained using first principle theoretical calculations, and establishes a practical route for in situ control of quantum anomalous Hall states and topology.

Confinement-Induced Chiral Edge Channel Interaction in Quantum Anomalous Hall Insulators

In quantum anomalous Hall (QAH) insulators, the interior is insulating but electrons can travel with zero resistance along one-dimensional (1D) conducting paths known as chiral edge channels (CECs). These CECs have been predicted to be confined to the 1D edges and exponentially decay in the two-dimensional (2D) bulk. In this Letter, we present the results of a systematic study of QAH devices fashioned in a Hall bar geometry of different widths under gate voltages. At the charge neutral point, the QAH effect persists in a Hall bar device with a width of only ∼72  nm, implying the intrinsic decaying length of CECs is less than ∼36  nm. In the electron-doped regime, we find that the Hall resistance deviates quickly from the quantized value when the sample width is less than 1  μm. Our theoretical calculations suggest that the wave function of CEC first decays exponentially and then shows a long tail due to disorder-induced bulk states. Therefore, the deviation from the quantized Hall resistance in narrow QAH samples originates from the interaction between two opposite CECs mediated by disorder-induced bulk states in QAH insulators, consistent with our experimental observations.

Creation of chiral interface channels for quantized transport in magnetic topological insulator multilayer heterostructures

One-dimensional chiral interface channels can be created at the boundary of two quantum anomalous Hall (QAH) insulators with different Chern numbers. Such a QAH junction may function as a chiral edge current distributer at zero magnetic field, but its realization remains challenging. Here, by employing an in-situ mechanical mask, we use molecular beam epitaxy to synthesize QAH insulator junctions, in which two QAH insulators with different Chern numbers are connected along a one-dimensional junction. For the junction between Chern numbers of 1 and -1, we observe quantized transport and demonstrate the appearance of the two parallel propagating chiral interface channels along the magnetic domain wall at zero magnetic field. For the junction between Chern numbers of 1 and 2, our quantized transport shows that a single chiral interface channel appears at the interface. Our work lays the foundation for the development of QAH insulator-based electronic and spintronic devices and topological chiral networks.

Environmental Doping-Induced Degradation of the Quantum Anomalous Hall Insulators

The quantum anomalous Hall (QAH) insulator carries dissipation-free chiral edge current and thus provides a unique opportunity to develop energy-efficient transformative information technology. Despite promising advances, the QAH insulator has thus far eluded any practical applications. In addition to its low working temperature, the QAH state in magnetically doped topological insulators usually deteriorates with time in ambient conditions. In this work, we store three QAH devices with similar initial properties in different environments. The QAH device without a protection layer in air shows clear degradation and becomes hole-doped. The QAH device kept in an argon glovebox without a protection layer shows no measurable degradation after 560 h, and the device protected by a 3 nm AlO_x protection layer in air shows minimal degradation with stable QAH properties. Our work shows a route to preserve the dissipation-free chiral edge state in QAH devices for potential applications in quantum information technology.

Controllable Chirality and Band Gap of Quantum Anomalous Hall Insulators

Finding guiding principles to optimize properties of quantum anomalous Hall (QAH) insulators is of pivotal importance to fundamental science and applications. Here, we build a first-principles QAH material database of chirality and band gap, explore microscopic mechanisms determining the QAH material properties, and obtain a general physical picture that can help researchers comprehensively understand the QAH data. Our results reveal that the usually neglected Coulomb exchange is unexpectedly strong in a large class of QAH materials, which is the key to resolve experimental puzzles. Moreover, we identify simple indicators for property evaluation and suggest material design strategies to control QAH chirality and gap by tuning cooperative or competing contributions via magnetic codoping, heterostructuring, spin-orbit proximity, etc. The work is valuable to future research of magnetic topological physics and materials.

Direct visualization of edge state in even-layer MnBi2Te4 at zero magnetic field

Being the first intrinsic antiferromagnetic (AFM) topological insulator (TI), MnBi2Te4 is argued to be a topological axion state in its even-layer form due to the antiparallel magnetization between the top and bottom layers. Here we combine both transport and scanning microwave impedance microscopy (sMIM) to investigate such axion state in atomically thin MnBi2Te4 with even-layer thickness at zero magnetic field. While transport measurements show a zero Hall plateau signaturing the axion state, sMIM uncovers an unexpected edge state raising questions regarding the nature of the "axion state". Based on our model calculation, we propose that the edge state of even-layer MnBi2Te4 at zero field is derived from gapped helical edge states of the quantum spin Hall effect with time-reversal-symmetry breaking, when a crossover from a three-dimensional TI MnBi2Te4 to a two-dimensional TI occurs. Our finding thus signifies the richness of topological phases in MnB2Te4 that has yet to be fully explored.

Measured Potential Profile in a Quantum Anomalous Hall System Suggests Bulk-Dominated Current Flow

Ideally, quantum anomalous Hall systems should display zero longitudinal resistance. Yet in experimental quantum anomalous Hall systems elevated temperature can make the longitudinal resistance finite, indicating dissipative flow of electrons. Here, we show that the measured potentials at multiple locations within a device at elevated temperature are well described by solution of Laplace's equation, assuming spatially uniform conductivity, suggesting nonequilibrium current flows through the two-dimensional bulk. Extrapolation suggests that at even lower temperatures current may still flow primarily through the bulk rather than, as had been assumed, through edge modes. An argument for bulk current flow previously applied to quantum Hall systems supports this picture.

Probing the mesoscopic size limit of quantum anomalous Hall insulators

The inelastic scattering length (L_s) is a length scale of fundamental importance in condensed matters due to the relationship between inelastic scattering and quantum dephasing. In quantum anomalous Hall (QAH) materials, the mesoscopic length scale L_s plays an instrumental role in determining transport properties. Here we examine L_s in three regimes of the QAH system with distinct transport behaviors: the QAH, quantum critical, and insulating regimes. Although the resistance changes by five orders of magnitude when tuning between these distinct electronic phases, scaling analyses indicate a universal L_s among all regimes. Finally, mesoscopic scaled devices with sizes on the order of L_s were fabricated, enabling the direct detection of the value of L_s in QAH samples. Our results unveil the fundamental length scale that governs the transport behavior of QAH materials.

In quantum anomalous Hall (QAH) materials, the mesoscopic scattering length (L_s) plays an instrumental role in determining transport properties. Here, the authors examine L_s in three regimes (QAH, quantum critical, and insulating) with distinct transport behaviours, and find a universal L_s across all regimes.

Spontaneous Ferromagnetism Induced Topological Transition in EuB_{6}

The interplay between various symmetries and electronic bands topology is one of the core issues for topological quantum materials. Spontaneous magnetism, which leads to the breaking of time-reversal symmetry, has been proven to be a powerful approach to trigger various exotic topological phases. In this Letter, utilizing the combination of angle-resolved photoemission spectroscopy, magneto-optical Kerr effect microscopy, and first-principles calculations, we present the direct evidence on the realization of the long-sought spontaneous ferromagnetism induced topological transition in soft ferromagnetic EuB_{6}. Explicitly, we reveal the topological transition is from Z_{2}=1 topological insulator in paramagnetic state to χ=1 magnetic topological semimetal in low temperature ferromagnetic state. Our results demonstrate that the simple band structure near the Fermi level and rich topological phases make EuB_{6} an ideal platform to study the topological phase physics.

Physical Vapor Deposition Features of Ultrathin Nanocrystals of Bi2(TexSe1-x)3

Structural and electronic properties of ultrathin nanocrystals of chalcogenide Bi₂(Te_x Se_1-x)₃ were studied. The nanocrystals were formed from the parent compound Bi₂Te₂Se on as-grown and thermally oxidized Si(100) substrates using Ar-assisted physical vapor deposition, resulting in well-faceted single crystals several quintuple layers thick and a few hundreds nanometers large. The chemical composition and structure of the nanocrystals were analyzed by energy-dispersive X-ray spectroscopy, X-ray photoelectron spectroscopy, electron backscattering, and X-ray diffraction. The electron transport through nanocrystals connected to superconducting Nb electrodes demonstrated Josephson behavior, with the predominance of the topological channels [Stolyarov et al. Commun. Mater., 2020, 1, 38]. The present paper focuses on the effect of the growth conditions on the morphology, structural, and electronic properties of nanocrystals.

Direct Probe of the Ferromagnetic Proximity Effect at the Interface of SnTe/Fe Heterostructure by Polarized Neutron Reflectometry

Introducing magnetic order into a topological insulator (TI) system has attracted much attention with an expectation of realizing exotic phenomena such as the quantum anomalous Hall effect (QAHE) and axion insulator states. The magnetic proximity effect (MPE) is one of the promising schemes to induce the magnetic order on the surface of a TI without introducing disorder accompanied by doping magnetic impurities in the TI. In this study, we investigate the MPE at the interface of a heterostructure consisting of the topological crystalline insulator (TCI) SnTe and Fe by employing polarized neutron reflectometry. The ferromagnetic order penetrates ∼2.2 nm deep into the SnTe layer from the interface with Fe, which persists up to room temperature. This is induced by the MPE on the surface of the TCI preserving the coherent topological states without introducing the disorder by doping magnetic impurities. This would open up a way for realizing next-generation spintronics and quantum computational devices.

Quantum Imaging of Magnetic Phase Transitions and Spin Fluctuations in Intrinsic Magnetic Topological Nanoflakes

Topological materials featuring exotic band structures, unconventional current flow patterns, and emergent organizing principles offer attractive platforms for the development of next-generation transformative quantum electronic technologies. The family of MnBi₂Te₄ (Bi₂Te₃)_n materials is naturally relevant in this context due to their nontrivial band topology, tunable magnetism, and recently discovered extraordinary quantum transport behaviors. Despite numerous pioneering studies to date, the local magnetic properties of MnBi₂Te₄ (Bi₂Te₃)_n remain an open question, hindering a comprehensive understanding of their fundamental material properties. Exploiting nitrogen-vacancy (NV) centers in diamond, we report nanoscale quantum imaging of the magnetic phase transitions and spin fluctuations in exfoliated MnBi₄Te₇ flakes, revealing the underlying spin transport physics and magnetic domains at the nanoscale. Our results highlight the unique advantage of NV centers in exploring the magnetic properties of emergent quantum materials, opening new opportunities for investigating the interplay between topology and magnetism.

Zero Magnetic Field Plateau Phase Transition in Higher Chern Number Quantum Anomalous Hall Insulators

The plateau-to-plateau transition in quantum Hall effect under high magnetic fields is a celebrated quantum phase transition between two topological states. It can be achieved by either sweeping the magnetic field or tuning the carrier density. The recent realization of the quantum anomalous Hall (QAH) insulators with tunable Chern numbers introduces the channel degree of freedom to the dissipation-free chiral edge transport and makes the study of the quantum phase transition between two topological states under zero magnetic field possible. Here, we synthesized the magnetic topological insulator (TI)/TI pentalayer heterostructures with different Cr doping concentrations in the middle magnetic TI layers using molecular beam epitaxy. By performing transport measurements, we found a potential plateau phase transition between C=1 and C=2 QAH states under zero magnetic field. In tuning the transition, the Hall resistance monotonically decreases from h/e^{2} to h/2e^{2}, concurrently, the longitudinal resistance exhibits a maximum at the critical point. Our results show that the ratio between the Hall resistance and the longitudinal resistance is greater than 1 at the critical point, which indicates that the original chiral edge channel from the C=1 QAH state coexists with the dissipative bulk conduction channels. Subsequently, these bulk conduction channels appear to self-organize and form the second chiral edge channel in completing the plateau phase transition. Our study will motivate further investigations of this novel Chern number change-induced quantum phase transition and advance the development of the QAH chiral edge current-based electronic and spintronic devices.

Mesoscopic Transport of Quantum Anomalous Hall Effect in the Submicron Size Regime

The quantum anomalous Hall (QAH) effect has been demonstrated in two-dimensional topological insulator systems incorporated with ferromagnetism. However, a comprehensive understanding of mesoscopic transport in submicron QAH devices has not yet been established. Here we fabricated miniaturized QAH devices with channel widths down to 600 nm, where the QAH features are still preserved. A backscattering channel is formed in narrow QAH devices through percolative hopping between 2D compressible puddles. Large resistance fluctuations are observed in narrow devices near the coercive field, which is associated with collective interference between intersecting paths along domain walls when the device geometry is smaller than the phase coherence length L_{ϕ}. Through measurement of size-dependent breakdown current, we confirmed that the chiral edge states are confined at the physical boundary with its width on the order of Fermi wavelength.

Floquet topological phases with high Chern numbers in a periodically driven extended Su-Schrieffer-Heeger model

The high Chern number phases with the Chern number |C| > 1 are observed in this study of a periodically driven extended Su-Schrieffer-Heeger (E-SSH) model with a cyclic parameter. Besides the standard intra-dimer and the nearest-neighbor inter-dimer hopping of the SSH model, an additional next-nearest-neighbor hopping is considered in the E-SSH model. The cyclic parameter, which plays the role of a synthetic dimension, is invoked as a modulation of the hopping strengths. A rigorous analysis of different phase diagrams has shown multiple Floquet topological phase transitions among the high Chern number phases. These phase transitions can be controlled by the strength and frequency of the periodic driving. Instead of applying perturbation theory, the whole analysis is done by Floquet replica technique. This gives a freedom to study high as well as low-frequency effects on the system by considering less or more number of photon sectors. This system can be experimentally realized through a pulse sequence scheme in the optical lattice setup.

Phase transitions in intrinsic magnetic topological insulator with high-frequency pumping

In this work, we investigate the topological phase transitions in an effective model for a topological thin film with high-frequency pumping. In particular, our results show that the circularly polarized light can break the time-reversal symmetry and induce the quantum anomalous Hall insulator (QAHI) phase. Meanwhile, the bulk magnetic moment can also break the time-reversal symmetry. Therefore, it shows rich phase diagram by tuning the intensity of the light and the thickness of the thin film. Using the parameters fitted by experimental data, we give the topological phase diagram of the Cr-doped Bi₂Se₃thin film, showing that by modulating the strength of the polarized optical field in an experimentally accessible range, there are four different phases: the normal insulator phase, the time-reversal-symmetry-broken quantum spin Hall insulator phase, and two different QAHI phases with opposite Chern numbers. Comparing with the non-doped Bi₂Se₃, it is found that the interplay between the light and bulk magnetic moment separates the two different QAHI phases with opposite Chern numbers. The results show that an intrinsic magnetic topological insulator with high-frequency pumping is an ideal platform for further exploring various topological phenomena with a spontaneously broken time-reversal symmetry.

Enormous Berry-Curvature-Based Anomalous Hall Effect in Topological Insulator (Bi,Sb)2Te3 on Ferrimagnetic Europium Iron Garnet beyond 400 K

To realize the quantum anomalous Hall effect (QAHE) at elevated temperatures, the approach of magnetic proximity effect (MPE) was adopted to break the time-reversal symmetry in the topological insulator (Bi_0.3Sb_0.7)₂Te₃ (BST) based heterostructures with a ferrimagnetic insulator europium iron garnet (EuIG) of perpendicular magnetic anisotropy. Here we demonstrate large anomalous Hall resistance (R_AHE) exceeding 8 Ω (ρ_AHE of 3.2 μΩ·cm) at 300 K and sustaining to 400 K in 35 BST/EuIG samples, surpassing the past record of 0.28 Ω (ρ_AHE of 0.14 μΩ·cm) at 300 K. The large R_AHE is attributed to an atomically abrupt, Fe-rich interface between BST and EuIG. Importantly, the gate dependence of the AHE loops shows no sign change with varying chemical potential. This observation is supported by our first-principles calculations via applying a gradient Zeeman field plus a contact potential on BST. Our calculations further demonstrate that the AHE in this heterostructure is attributed to the intrinsic Berry curvature. Furthermore, for gate-biased 4 nm BST on EuIG, a pronounced topological Hall effect-like (THE-like) feature coexisting with AHE is observed at the negative top-gate voltage up to 15 K. Interface tuning with theoretical calculations has realized topologically distinct phenomena in tailored magnetic TI-based heterostructures.

Thickness-Driven Quantum Anomalous Hall Phase Transition in Magnetic Topological Insulator Thin Films

The quantized version of the anomalous Hall effect realized in magnetic topological insulators (MTIs) has great potential for the development of topological quantum physics and low-power electronic/spintronic applications. Here we report the thickness-tailored quantum anomalous Hall (QAH) effect in Cr-doped (Bi,Sb)₂Te₃ thin films by tuning the system across the two-dimensional (2D) limit. In addition to the Chern number-related QAH phase transition, we also demonstrate that the induced hybridization gap plays an indispensable role in determining the ground magnetic state of the MTIs; namely, the spontaneous magnetization owing to considerable Van Vleck spin susceptibility guarantees the zero-field QAH state with unitary scaling law in thick samples, while the quantization of the Hall conductance can only be achieved with the assistance of external magnetic fields in ultrathin films. The modulation of topology and magnetism through structural engineering may provide useful guidance for the pursuit of other QAH-based phase diagrams and functionalities.

Topological spintronics and magnetoelectronics

Topological electronic materials, such as topological insulators, are distinct from trivial materials in the topology of their electronic band structures that lead to robust, unconventional topological states, which could bring revolutionary developments in electronics. This Perspective summarizes developments of topological insulators in various electronic applications including spintronics and magnetoelectronics. We group and analyse several important phenomena in spintronics using topological insulators, including spin-orbit torque, the magnetic proximity effect, interplay between antiferromagnetism and topology, and the formation of topological spin textures. We also outline recent developments in magnetoelectronics such as the axion insulator and the topological magnetoelectric effect observed using different topological insulators.